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Martin Sleziak
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Denis Serre
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Serre's A Course in Arithmetic gives essentially the following proof of the three-squares theorem, which says that an integer a$a$ is the sum of three squares if and only if it is not of the form 4^m (8n + 7)$4^m (8n + 7)$ : first one shows that the condition is necessary, which is straightforward. To show it is sufficient, a lemma of Davenport and Cassels, using Hasse-Minkowski, shows that a$a$ is the sum of three rational squares. Then something magical happens:

Let C$C$ denote the circle x^2 + y^2 + z^2 = a$x^2 + y^2 + z^2 = a$. We are given a rational point p$p$ on this circle. Round the coordinates of p$p$ to the closest integer point q$q$, then draw the line through p$p$ and q$q$, which intersects C$C$ at a rational point p'$p'$. Round the coordinates of p'$p'$ to the closest integer point q'$q'$, and repeat this process. A straightforward calculation shows that the least common multiple of the denominators of the points p'$p'$, p''$p''$, ... are strictly decreasing, so this process terminates at an integer point on C$C$.

Bjorn Poonen, after presenting this proof in class, remarked that he had no intuition for why this should work. Does anyone have a reply?

Edit: Let me suggest a possible reformulation of the question as follows. Complete the analogy: Hensel's lemma is to Newton's method as this technique is to _____________________.

Serre's A Course in Arithmetic gives essentially the following proof of the three-squares theorem, which says that an integer a is the sum of three squares if and only if it is not of the form 4^m (8n + 7): first one shows that the condition is necessary, which is straightforward. To show it is sufficient, a lemma of Davenport and Cassels, using Hasse-Minkowski, shows that a is the sum of three rational squares. Then something magical happens:

Let C denote the circle x^2 + y^2 + z^2 = a. We are given a rational point p on this circle. Round the coordinates of p to the closest integer point q, then draw the line through p and q, which intersects C at a rational point p'. Round the coordinates of p' to the closest integer point q', and repeat this process. A straightforward calculation shows that the least common multiple of the denominators of the points p', p'', ... are strictly decreasing, so this process terminates at an integer point on C.

Bjorn Poonen, after presenting this proof in class, remarked that he had no intuition for why this should work. Does anyone have a reply?

Edit: Let me suggest a possible reformulation of the question as follows. Complete the analogy: Hensel's lemma is to Newton's method as this technique is to _____________________.

Serre's A Course in Arithmetic gives essentially the following proof of the three-squares theorem, which says that an integer $a$ is the sum of three squares if and only if it is not of the form $4^m (8n + 7)$ : first one shows that the condition is necessary, which is straightforward. To show it is sufficient, a lemma of Davenport and Cassels, using Hasse-Minkowski, shows that $a$ is the sum of three rational squares. Then something magical happens:

Let $C$ denote the circle $x^2 + y^2 + z^2 = a$. We are given a rational point $p$ on this circle. Round the coordinates of $p$ to the closest integer point $q$, then draw the line through $p$ and $q$, which intersects $C$ at a rational point $p'$. Round the coordinates of $p'$ to the closest integer point $q'$, and repeat this process. A straightforward calculation shows that the least common multiple of the denominators of the points $p'$, $p''$, ... are strictly decreasing, so this process terminates at an integer point on $C$.

Bjorn Poonen, after presenting this proof in class, remarked that he had no intuition for why this should work. Does anyone have a reply?

Edit: Let me suggest a possible reformulation of the question as follows. Complete the analogy: Hensel's lemma is to Newton's method as this technique is to _____________________.

Reformulated question.
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Qiaochu Yuan
Qiaochu Yuan
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Serre's A Course in Arithmetic gives essentially the following proof of the three-squares theorem, which says that an integer a is the sum of three squares if and only if it is not of the form 4^m (8n + 7): first one shows that the condition is necessary, which is straightforward. To show it is sufficient, a lemma of Davenport and Cassels, using Hasse-Minkowski, shows that a is the sum of three rational squares. Then something magical happens:

Let C denote the circle x^2 + y^2 + z^2 = a. We are given a rational point p on this circle. Round the coordinates of p to the closest integer point q, then draw the line through p and q, which intersects C at a rational point p'. Round the coordinates of p' to the closest integer point q', and repeat this process. A straightforward calculation shows that the least common multiple of the denominators of the points p', p'', ... are strictly decreasing, so this process terminates at an integer point on C.

Bjorn Poonen, after presenting this proof in class, remarked that he had no intuition for why this should work. Does anyone have a reply?

Edit: Let me suggest a possible reformulation of the question as follows. Complete the analogy: Hensel's lemma is to Newton's method as this technique is to _____________________.

Serre's A Course in Arithmetic gives essentially the following proof of the three-squares theorem, which says that an integer a is the sum of three squares if and only if it is not of the form 4^m (8n + 7): first one shows that the condition is necessary, which is straightforward. To show it is sufficient, a lemma of Davenport and Cassels, using Hasse-Minkowski, shows that a is the sum of three rational squares. Then something magical happens:

Let C denote the circle x^2 + y^2 + z^2 = a. We are given a rational point p on this circle. Round the coordinates of p to the closest integer point q, then draw the line through p and q, which intersects C at a rational point p'. Round the coordinates of p' to the closest integer point q', and repeat this process. A straightforward calculation shows that the least common multiple of the denominators of the points p', p'', ... are strictly decreasing, so this process terminates at an integer point on C.

Bjorn Poonen, after presenting this proof in class, remarked that he had no intuition for why this should work. Does anyone have a reply?

Serre's A Course in Arithmetic gives essentially the following proof of the three-squares theorem, which says that an integer a is the sum of three squares if and only if it is not of the form 4^m (8n + 7): first one shows that the condition is necessary, which is straightforward. To show it is sufficient, a lemma of Davenport and Cassels, using Hasse-Minkowski, shows that a is the sum of three rational squares. Then something magical happens:

Let C denote the circle x^2 + y^2 + z^2 = a. We are given a rational point p on this circle. Round the coordinates of p to the closest integer point q, then draw the line through p and q, which intersects C at a rational point p'. Round the coordinates of p' to the closest integer point q', and repeat this process. A straightforward calculation shows that the least common multiple of the denominators of the points p', p'', ... are strictly decreasing, so this process terminates at an integer point on C.

Bjorn Poonen, after presenting this proof in class, remarked that he had no intuition for why this should work. Does anyone have a reply?

Edit: Let me suggest a possible reformulation of the question as follows. Complete the analogy: Hensel's lemma is to Newton's method as this technique is to _____________________.

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Qiaochu Yuan
Qiaochu Yuan
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